Customers arrive at a holding facility at random according to a Poison process (Nt , t ≥ 0) having rate λ > 0. The facility processes in batches of size K. That is, the first K − 1 customers wait until the arrival of the K-th customer. Then all are passed simultaneously, and the process repeats. Service times are simultaneous. (i) Find the expected valued of the first dispatched time T given that at the beginning there are no customers. (ii) Show that E[∫0tN(t)dt]= K(K − 1)/2λ.

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